A general construction of barycentric coordinates over convex polygons

Barycentric coordinates are unique for triangles, but there are many possible generalizations to arbitrary convex polygons. in this paper we derive sharp upper and lower bounds and use them to show that all barycentric coordinates are identical at the boundary of the polygon. we then present a general approach for constructing such coordinates and use it to show that the wachspress, mean value, and discrete harmonic coordinates all belong to a unifying one-parameter family. but the only members of this family that are positive are the wachspress and mean value coordinates. however, our general approach does allow us to construct several new sets of barycentric coordinates.